Dear Ed, the smallest number whose sum of divisors is 10 times the number is: 487# * 29# * 10! / 2 which is about 1.19630286446804*10^217 and its factor is 10.00036 (Lejeunes number is about: 2.29353535102896E+222 and its factor is 10.0129) I have also found the samllest numbers which exceed the factor 1,2,3,4,5,6,7,8,9,10. This gives the sequence A023199 (which - alas - is already known) I enjoyed also your mentioning of DROD, which I did not know so far. I have bought it and am now trying to solve some of its riddles - thats really a labyrinth of dungeons.... Thanks Yours - Bodo Zinser from Munich ----------------------------------------- I stop now, this one is the smallest I found (and I supposed to be the smallest at all) 1196302864468037060143273209658450423930161241757076738827918396950538234505 9016648535347468836046455234043935194500296748473838972491647043636698551317 157043820830462348056767777839861694764691622417511967632039040000 = 487# * 29# * 10! / 2 (=10.0004 times sum of it's divisor) May be you can wrote about it on your page http://www.mathpuzzle.com/ Thanks, Vincent Lejeune ----------------------------------------------- Smallest n : sigma0(n) / n >= 11 that I found was 2^10 * 3^6 * 5^4 * 7^3 * ( 11 ... 29 )^2 * (31 ... 487) or 487# 29# 10! / 2 Regards, Brian Trial ------------------------------------------------